Linear programming interpretations of Mather's variational principle
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 693-702.

We discuss some implications of linear programming for Mather theory [13, 14, 15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an n-dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [6, 7, 8, 5].

DOI : 10.1051/cocv:2002030
Classification : 90C05, 35F20
Mots-clés : linear programming, duality, weak KAM theory 
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     title = {Linear programming interpretations of {Mather's} variational principle},
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Evans, L. C.; Gomes, D. Linear programming interpretations of Mather's variational principle. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 693-702. doi : 10.1051/cocv:2002030. http://www.numdam.org/articles/10.1051/cocv:2002030/

[1] E.J. Anderson and P. Nash, Linear Programming in Infinite Dimensional Spaces. Wiley (1987). | MR | Zbl

[2] D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific (1997). | Zbl

[3] L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer (survey paper). Available at the website of LCE, at math.berkeley.edu | Zbl

[4] L.C. Evans, Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations (to appear). | MR | Zbl

[5] L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I. Arch. Rational Mech. Anal. 157 (2001) 1-33. | MR | Zbl

[6] A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1043-1046. | MR | Zbl

[7] A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 649-652. | MR | Zbl

[8] A. Fathi, Weak KAM theory in Lagrangian Dynamics, Preliminary Version. Lecture Notes (2001).

[9] J. Franklin, Methods of Mathematical Economics. SIAM, Classics in Appl. Math. 37 (2002). | MR | Zbl

[10] D. Gomes, Numerical methods and Hamilton-Jacobi equations (to appear).

[11] P. Lax, Linear Algebra. John Wiley (1997). | MR | Zbl

[12] P.-L. Lions, G. Papanicolaou and S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations. CIRCA (1988) (unpublished).

[13] J. Mather, Minimal measures. Comment. Math Helvetici 64 (1989) 375-394. | MR | Zbl

[14] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169-207. | MR | Zbl

[15] J. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems. Transition to Chaos in Classical and Quantum Mechanics, edited by S. Graffi. Sringer, Lecture Notes in Math. 1589 (1994). | MR | Zbl

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